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Hankel Operators on Fock Spaces and Related Bergman Kernel Estimates

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Abstract

Hankel operators with anti-holomorphic symbols are studied for a large class of weighted Fock spaces on ℂn. The weights defining these Hilbert spaces are radial and subject to a mild smoothness condition. In addition, it is assumed that the weights decay at least as fast as the classical Gaussian weight. The main result of the paper says that a Hankel operator on such a Fock space is bounded if and only if the symbol belongs to a certain BMOA space, defined via the Berezin transform. The latter space coincides with a corresponding Bloch space which is defined by means of the Bergman metric. This characterization of boundedness relies on certain precise estimates for the Bergman kernel and the Bergman metric. Characterizations of compact Hankel operators and Schatten class Hankel operators are also given. In the latter case, results on Carleson measures and Toeplitz operators along with Hörmander’s L 2 estimates for the \(\bar{\partial}\) operator are key ingredients in the proof.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), 7491, Trondheim, Norway

    Kristian Seip

  2. LATP, U.M.R. C.N.R.S. 6632, CMI, Université de Provence, 39 Rue F-Joliot-Curie, 13453, Marseille Cedex 13, France

    El Hassan Youssfi

Authors
  1. Kristian Seip
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  2. El Hassan Youssfi
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Correspondence to Kristian Seip.

Additional information

Communicated by Richard Rochberg.

The first author is supported by the Research Council of Norway grant 185359/V30. The second author is supported by the French ANR DYNOP, Blanc07-198398.

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Seip, K., Youssfi, E.H. Hankel Operators on Fock Spaces and Related Bergman Kernel Estimates. J Geom Anal 23, 170–201 (2013). https://doi.org/10.1007/s12220-011-9241-9

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  • DOI: https://doi.org/10.1007/s12220-011-9241-9

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